Regular models of hyperelliptic curves

We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. We also explain a powerful refinement of the Mordell--Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on a two genus 2 hyperelliptic curves with Mordell--Weil Jacobian ranks of 3 and 6.

The influence of changes in strength of exposed steel column bases (ESCBs) on the dynamic response of low-rise steel frames was studied by building on the theoretical sliding hysteresis curve model of ESCBs on the basis of a previous experimental study. The first-floor shear–weight ratio (CB = 0·30 and 0·35) and frame shape were taken into account in designing 16 strong-column/weak-beam regular low-rise steel frame models with the ratio of the strength of column bases to the strength of the first-floor column as the main study parameter. The results showed that when the strength coefficient of the steel column base was lower than 0·7, the model deformed at the first floor and plastic energy was concentrated if the first-floor shear–weight ratio was 0·30, while the first-floor deformation and plastic energy concentration of the model eased if the first-floor shear–weight ratio was 0·35. In addition, the lower the strength coefficient of the steel column base, the more obvious was the easing effect. The energy absorption of the column bases was not significant, but the column bases were able to transfer input seismic energy to other components during the plastic sliding process. The dynamic response hysteresis curves of the column bases matched the theoretical model well.

Score limitation at the top of a scale is commonly termed “ceiling effect.” Ceiling effects can lead to serious artifactual parameter estimates in most data analysis. This study examines the consequences of ceiling effects in longitudinal data analysis and investigates several methods of dealing with ceiling effects through Monte Carlo simulations and empirical data analyses. Data were simulated based on a latent growth curve model with T = 5 occasions. The proportion of the ceiling data [10%–40%] was manipulated by using different thresholds, and estimated parameters were examined for R = 500 replications. The results showed that ceiling effects led to incorrect model selection and biased parameter estimation (shape of the curve and magnitude of the changes) when regular growth curve models were applied. The Tobit growth curve model, instead, performed very well in dealing with ceiling effects in longitudinal data analysis. The Tobit growth curve model was then applied in an empirical cognitive aging study and the results were discussed.

The accuracy of cerebral blood flow (CBF) estimates from arterial spin labeling (ASL) is affected by the presence of both gray matter (GM) and white matter within any voxel. Recently a partial volume (PV) correction method for ASL has been demonstrated (Asllani et al. Magn Reson Med 2008; 60:1362-1371), where PV estimates were used with a local linear regression to separate the GM and white matter ASL signal. Here a new PV correction method for multi-inversion time ASL is proposed that exploits PV estimates within a spatially regularized kinetic curve model analysis. The proposed method exploits both PV estimates and the different kinetics of the ASL signal arising from GM and white matter. The new correction method is shown, on both simulated and real data, to provide correction of GM CBF comparable to a linear regression approach, whilst preserving greater spatial detail in the CBF image. On real data corrected GM CBF values were found to be largely independent of GM PV, implying that the correction had been successful. Increases of mean GM CBF after correction of 69-80% were observed.

Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym variety and determine the interesting simple factors other than elliptic curves or hyperelliptic curves with small genus which come up in $J_n$ so that the endomorphism rings contains the totally real field $\mathbb{Q}(\zeta_n+\zeta^{-1}_n)$.

The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field E to logarithmic derivatives of Artin L-functions at $$s=0$$ . In this paper, we prove that if F is any fixed totally real number field of degree $$[F:\mathbb {Q}] \ge 3$$ , then there are infinitely many effective, “positive density” sets of CM extensions E / F such that $$E/\mathbb {Q}$$ is non-abelian and the Colmez conjecture is true for E. Moreover, these CM extensions are explicitly constructed to be ramified at arbitrary prescribed sets of prime ideals of F. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be viewed as an explicit non-abelian Chowla–Selberg formula. Our results rely crucially on an averaged version of the Colmez conjecture which was recently proved independently by Andreatta–Goren–Howard–Madapusi Pera and Yuan–Zhang.

We associate a weight function to pairs consisting of a smooth and proper variety X over a complete discretely valued field and a differential form on X of maximal degree. This weight function is a real-valued function on the non-archimedean analytification of X. It is piecewise affine on the skeleton of any regular model with strict normal crossings of X, and strictly ascending as one moves away from the skeleton. We apply these properties to the study of the Kontsevich-Soibelman skeleton of such a pair, and we prove that this skeleton is connected when X has geometric genus one. This result can be viewed as an analog of the Shokurov-Kollar connectedness theorem in birational geometry.

Let W / K W/K be a nonempty scheme over the field of fractions of a Henselian local ring R R . A result of Gabber, Liu and Lorenzini [Invent. Math. 192 (2013), pp. 567–626] shows that the G C D GCD of the set of degrees of closed points on W W (which is called the index of W / K W/K ) can be computed from data pertaining only to the special fiber of a proper regular model of W W over R R . We show that the entire set of degrees of closed points on W W can be computed from data pertaining only to the special fiber, provided the special fiber is a strict normal crossings divisor. As a consequence we obtain an algorithm to compute the degree set of any smooth curve over a Henselian field with finite or algebraically closed residue field. Using this we show that degree sets of curves over such fields can be dramatically different than degree sets of curves over finitely generated fields. For example, while the degree set of a curve over a finitely generated field contains all sufficiently large multiples of the index, there are curves over p p -adic fields with index 1 1 whose degree set excludes all integers that are coprime to 6 6 .